## Arithmetic and Algebra

Something we all ‘learn’ at school, and thereafter take for granted. It’s also where I typically make mistakes. I can work my way through some horrendous manipulation and calculation, only for the last step to be the ‘simple’ addition of two numbers, and that’s where I can so easily mess up. I think the reason is because I’m not thinking about it. Anything complicated and I’m visualising what’s happening, simple arithmetical operations or algebraic manipulations… Probably because they are just ‘rules’ that we learn: Rules that without thought are easy to misapply. The only solution? Practice. Don’t take things for granted just because they are ‘easy’. Study them and understand them just as you would anything else. Practice!

There’s the terminology:

If *t=a+b* then *t ≡ ***sum**

*t=a-b* *t ≡ ***difference**

*t=ab* *t ≡ ***product**, and *a* and *b ≡ ***factors**

*t=a/b* *t ≡ ***quotient**, and *b* ≡ **divisor
**

You need to notice that the ordering of terms is immaterial if they are being added or multiplied, but not if they are being subtracted or divided: i.e.

*ab=ba*

* a+b=b+a*

* a/b≠b/a*

* a-b≠b-a*

The Conventions: When you evaluate an expression you do so in the order Brackets (what’s within), Powers (‘of’), Division, Multiplication, Addition, Subtraction i.e. **BODMAS**.

Some things worth remembering:

* (a+b) ^{2} = a^{2}+2ab+b^{2}*

* (a-b) ^{ 2}=a^{2}-2ab+b^{2
}*

^{ }(a+b)(a-b)=a^{2}-b^{2
}

And then the rules:

* a/b *×* c/d = ac/bd
*

* a/b / c/d = ad/bc
*

* a/b + c/d = (ad+cb)/bd
*

* a/b – c/d = (ad-cb)/bd
*

Powers:

* a ^{m} *

*a*

^{n}= a^{m+n}* (a ^{m})^{n} = a^{mn}*

* a ^{1}= a
*

* a ^{0}=1
*

* a ^{-1}=1/a
*

* a ^{-2}=1/a^{2}*

^{ n}√a=a^{1/n
}

Thereafter, it’s just practice. Practice those algebraic manipulations until you can do them in your head 😉